McCaskill, who had been living with dementia, died on Saturday his daughter Kirsty told the corporation in a statement.
The statement said: “Ian McCaskill’s family is sad to announce that Ian, 78, died on Saturday 10 December 2016 after living with dementia for the past five years.
“Ian was a truly lovely man who loved his family unconditionally and brought lots of sunshine to people’s lives with his friendly smile, kindness and sharp wit.
“He is survived by his wife Pat, whom he adored, two daughters, Vicky and Kirsty, two step-sons, Tim and Matthew, and nine grandchildren. He will be deeply missed.”
Known for his Scottish accent, McCaskill presented the weather on the BBC from 1978 to 1998.
The Glasgow-born star retired in 1998 and appeared on several television shows, including Celebrity Fit Club in 2002.
McCaskill became a meteorologist in the RAF and later moved to the Met Office.
He began working at the BBC at a time when it was not common for presenters to have regional accents, but went on to become popular with TV audiences.
He was once voted Britain’s sexiest weather presenter and even had his own Spitting Image puppet, which he said in an interview was “the greatest compliment”.
Weatherman Liam Dutton was among those to pay tribute to McCaskill on Twitter.
He wrote: “Sad news about Ian McCaskill - a BBC weatherman I grew up watching. He was a lovely guy with a good sense of humour.”
Weather presenter Paul Hudson, who co-wrote the book Frozen In Time with McCaskill, said: “Really sorry to hear Ian McCaskill has passed away.
“We had great fun writing our book Frozen In Time, a warm, funny, generous & kind man.”
McCaskill once said he had some bad moments presenting the weather.
In an interview with The Scotsman in 2006, he said: “On one occasion I was asked - or told, really - by one of our senior forecasters to forecast four inches of snow for the south-east of England.
“So I promised every child in the south-east they would wake up to snow. And of course, there wasn’t a flake.”